Herein, a finite strain microscale model for magnetorheological elastomers (MREs) based on a general continuum formulation of the magneto-mechanical boundary value problem is introduced. The modelling approach enables to consider particles with magnetically soft or magnetically hard behavior. In order to connect the microscopic fields to effective macroscopic quantities, a suitable computational homogenization scheme is used. The microstructure of the considered MRE is discretized and the problem is solved numerically in terms of a coupled nonlinear finite element approach. Using the presented framework, the influence of the particle shape on the magnetostrictive effect of MREs filled with magnetically soft particles is discussed. Furthermore, the effective macroscopic hystereses of an MRE filled with magnetically hard particles are calculated.